Any common factor greater than $1$ would qualify as a possible value of $k$, but since the statement of problem implies that the answer is unique, there better be only one such factor (else there would be more than one possible answer). Using that knowledge, you can expect that $k$ is prime. Since $2312-1417$ is odd, $k$ must be odd. Since $1059 - 701= 358 = 2\cdot 179$, and since $179$ is prime, $k$ must be 179. Of course it's worth checking (just to be sure) that $179$ divides all three differences.

To find $r$, find the remainder when any of the original numbers is divided by $179$ (might as well use the smallest $701$).

As I described in my previous reply, $n$ can be any common factor of the differences between successive pairs (e.g., the gaps). Thus for the 4 term sequence $17, 44, 65, 89$ that you give above, there are 3 gaps, namely $27, 21, 24$. The gcd of those 3 numbers is $3$. Any positive integer which divides the gcd would also give a valid $n$. If we explicitly exclude $n = 1$ (which would always work), then in this case, the only valid $n$ is $n = 3$.